What does it mean for two conjectures to be incompatible?

Question

What does it mean for two conjectures to be incompatible?

I read about Incompatibility of two Hardy-Littlewood Conjectures.

http://mathworld.wolfram.com/Hardy-LittlewoodConjectures.html

What does it mean?

Answer 1

It means that once you prove either of the conjectures to be true, you've automatically proven the other one to be false.